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Two-Numbers and Their Applications-A Survey Bull. Belg. Math. Soc. Simon Stevin 25 (2018), no. 4, pages xxx–xxx TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY BANG-YEN CHEN In memory of Professor Tadashi Nagano (1930 - 2017) Abstract. The notion of two-numbers of connected Riemannian man- ifolds was introduced about 35 years ago in [Un invariant geometrique riemannien, C. R. Acad. Sci. Paris Math. 295 (1982), 389–391] by B.-Y. Chen and T. Nagano. Later, two-numbers have been studied by a number of mathematicians and it was then proved that two-numbers related closely with several important areas in mathematics. The main purpose of this article is to survey on two-numbers and their applica- tions. At the end of this survey, we present several open problems and conjectures on two-numbers. 1. two-numbers and maximal antipodal sets The notion of the two-numbers of connected Riemannian manifolds was introduced about 35 years ago in [22]. The primeval concept of two-numbers is the notion of antipodal points on a circle. For a circle S1 in the Euclidean plane R2, the antipodal point q of a point p S1 is the point in S1 which is diametrically opposite to it. ∈ A geodesic on a Riemannian manifold M is a smooth curve which yields locally the shortest distance between any two nearby points. Because every arXiv:1805.04796v1 [math.DG] 12 May 2018 closed geodesic in a Riemannian manifold M is isometric to a circle S1, antipodal points can be defined for closed geodesics in M. More precisely, a point q in a closed geodesic is called an antipodal point of another point p 2010 Mathematics Subject Classification. Primary 00-02; Secondary 14-02, 22-02, 53- 02, 58-02. Key words and phrases. Two-number, (M+, M−)-theory, 2-rank, compact symmet- ric space, homotopy, homology, antipodal set, Betti number, Euler number, arithmetic distance. This is the detailed version of my plenary talk “Two-numbers and their applications” at the International Conference “Pure and Applied Differential Geometry - PADGE 2017” held at KU Leuven, Belgium, from August 21 till August 25, 2017. 1 2 B.-Y. CHEN on the same closed geodesic if the distance d(p,q) between p and q on the two arcs connecting p and q are equal. In this article, a closed geodesic in a Riemannian manifold is also called a circle. A subset S of a Riemannian manifold M is called an antipodal set if any two points in S are antipodal on some circle of M connecting them. An antipodal set A2M in a connected Riemannian manifold M is called a maximal antipodal set if it doesn’t lie in any antipodal set as a proper subset. The supremum of the cardinality of all maximal antipodal set of M is called the two-number of M, simply denote by #2M. When a Riemannian manifold M contains no closed geodesics, e.g., a Euclidean space, we put #2M = 0. n n Obviously, we have #2S = 2 for any n-sphere S . On the other hand, we also have # M 2 (1.1) 2 ≥ for every compact connected Riemannian manifold M, because compact connected Riemannian manifolds contain at least one close geodesic (cf. [54]). Remark 1.1. A maximal antipodal set A2M of a Riemannian manifold M is also later known as a 2-set or a great set for short in some literatures. 2. Antipodal points in algebraic topology Besides in geometry, the notion of antipodal points plays some significant roles in many areas of mathematics, physics, and applied sciences. The following result from algebraic topology is well-known. The Borsuk–Ulam Antipodal Theorem. Every continuous function from an n-sphere Sn into the Euclidean k-space Ek with k n maps some ≤ pair of antipodal points to the same point. In other word, if f : Sn Ek is → continuous, then there exists an x Sn such that f( x)= f(x). ∈ − Clearly, Borsuk-Ulam’s theorem fails for k>n, because Sn embedded in En+1. The Borsuk-Ulam theorem has numerous applications; range from combinatorics to differential equations and even economics. For n = 1, Borsuk–Ulam’s theorem implies that that at any moment there always exist a pair of opposite points on the earth’s equator with the same temperature. This assumes the temperature varies continuously. For n = 2, Borsuk–Ulam’s theorem implies that there is always a pair of antipodal points on the Earth’s surface with equal temperatures and equal barometric pressures, at any moment. TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 3 The Borsuk–Ulam theorem was conjectured by S. Ulam at the Scottish Caf´ein Lvov, Ukraine. Ulam’s conjecture was solved in 1933 by K. Borsuk [8]. It turned out that the result had been proved three years before in [55] by L. Lusternik and L. Schnirelmann (cf. [8, footnote, page 190]). Since then many alternative proofs and also many extensions of Borsuk–Ulam’s theorem have been found as collected by H. Steinlein in his survey article [74] including a list of 457 publications involving various generalizations of Borsuk-Ulam’s theorem. Another important theorem from algebraic topology with the same flavor is the following. Brouwer’s Fixed-Point Theorem. Every continuous function f : B B → from the unit n-ball B into itself has a fixed point. Brouwer’s fixed-point theorem has numerous applications to many fields as well. For instance, Brouwer’s fixed-point theorem and its extension by S. Kakutani in [47] (extending Brouwer’s fixed-point theorem to set-valued functions) played a central role in the proof of existence of general equilib- rium in Market Economies as developed in the 1950s by economics Nobel prize winners K. Arrow (1972) and G. Debreu (1983). Remark 2.1. In fact, Borsuk-Ulam’s antipodal theorem implies Brouwer’s fixed point theorem, see [75]. 3. Symmetric spaces The class of Riemannian manifolds with parallel Riemannian curvature tensor, i.e., R = 0, was first introduced independently by P. A. Shirokov ∇ in 1925 and by H. Levy in 1926. This class is known today as the class of locally symmetric Riemannian spaces (see, e.g., [16, page 292]). It was E.´ Cartan who noticed in 1926 that irreducible spaces of this type are separated into ten large classes each of which depends on one or two arbitrary integers, and in addition there exist twelve special classes corre- sponding to the exceptional simple groups. Based on this, E.´ Cartan created his theory of symmetric Riemannian spaces in his famous papers “Sur une classe remarquable d’espaces de Riemann” in 1926 [12]. Symmetric spaces are the most beautiful and important Riemannian man- ifolds. Such spaces arise in a wide variety of situations in both mathematics and physics. This class of spaces contains many prominent examples which are of great importance for various branches of mathematics, like compact Lie groups, Grassmannians and bounded symmetric domains. Symmetric 4 B.-Y. CHEN spaces are also important objects of study in representation theory, har- monic analysis as well as in differential geometry. An isometry s of a Riemannian manifold is called an involutive if s2 = id. A Riemannian manifold M is called a symmetric space if for each point x M there is an involutive isometry sx such that x is an isolated fixed ∈ point of sx; the involutive isometry sx = id is called the symmetry at x. 6 A Hermitian symmetric space is a Hermitian manifold M such that every point of M admits a symmetry preserving the Hermitian structure of M. Let M be a symmetric space. Denote by G the closure of the group of isometries on M generated by sp : p M in the compact-open topology. { ∈ } Then G is a Lie group which acts transitively on the symmetric space; hence the typical isotropy subgroup K, say at a point o M, is compact and ∈ M = G/K. Thus symmetric spaces are homogeneous spaces as well. From the point of view of Lie theory, a symmetric space is the quotient G/K of Lie group G by a Lie subgroup K, where the Lie algebra k of K is also required to be the (+1)-eigenspace of an involution of the Lie algebra g of G. In this paper, we will use standard symbols as in Helgason’s book [39] to denote symmetric spaces mostly. Here are a few minor exceptions. More specifically than AI, AI(n) denote SU(n)/SO(n), AII(n) := SU(2n)/Sp(n), n n n etc. Let Gd(R ), Gd(C ) and Gd(H ) denote the Grassmannians of d-dimen- sional subspaces in the real, complex and quaternion vector spaces (or mod- ules), respectively. The standard notations for the exceptional spaces such as G2, F4, E6, ..., GI, ..., EIX denote the simply-connected spaces where we write GI for G2/SO(4). We will denote by M ∗ the bottom space, i.e., the adjoint space in [39], of the space M. For a symmetric space M, the dimension of a maximal flat totally geodesic submanifold of M is a well-defined natural number; called the rank of M and denoted by rk(M). Clearly, the rank of a symmetric space is at least one. It is well-known that the class of rank one compact symmetric spaces consists of n-sphere Sn, a projective space FP n(F = R, C, H) and the 16-dimensional Cayley plane F II = OP 2 with O being the Cayley algebra. Obviously, every complete totally geodesic submanifold of a symmetric space is also a symmetric space. It follows from the equation of Gauss that rk(B) rk(M) (3.1) ≤ for every complete totally geodesic submanifold B of a symmetric space M (cf.
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